A New Sieve for the Twin Primes

Transcription

1 A ew Sieve for the Twin Primes and how the number of twin primes is related to the number of primes by H.L. Mitchell Department of mathematics CUY-The City College 160 Convent avenue ew York, Y USA hlmitchell88@yahoo.com

2 Abstract. We introduce a sieve for the number of twin primes less than n by sieving through the set {k Z + 6k < n}. We derive formula accordingly using the Euler product and the Brun Sieve. We then use the Prime umber Theorem and Mertens Theorem. The main results are: 1) A sieve for the twin primes similar to the sieve of Eratosthenes for primes involving only the values of k, the indices of the multiples of 6, ranging over k = p,5 p < n.it shows the uniform distribution of the pairs (6k-1,6k+1) that are not twin primes and the decreasing frequency of multiples of p as p increases. 2) A formula for the approximate number of twin primes less than in terms of the number of primes less than n 3) The asymptotic formula for the number of twin primes less than n verifying the Hardy Littlewood Conjecture Mathematics Subject Classification: 11A07; 11A41; 1105; 1125; 1136 Keywords and phrases: Primes, Twin Primes, Congruence, Sieve methods

3 1. Introduction The twin primes have been studied by a number of mathematicians over the past 3 centuries and thus far it is not known whether there exist infinitely many of them. Hardy and Littlewood proposed their famous conjecture in 1923 giving a formula for the number of twin primes less than a given integer n. We introduce a sieve for the twin primes less than n similar to the sieve of Eratosthenes for primes. It is applied twice to the set of all natural numbers k such that k < 6n and the range for the primes is p = 5 to p < n. We consider the set of all pairs (6k 1, 6k +1) which are less than n and delete the values of k such that 6k-1 is composite. This leaves us with the pairs for which 6k-1 is prime. From these we delete the values of k such that 6k+1 is composite and that leaves us with the twin primes less than. Using the Euler product formula, The Brun Sieve, The Prime number theorem and Mertens 3rd Theorem, we derive a formula for the approximation of π 2 (n) in terms of π (n) ( the number of primes less than n ) and the asymptotic formula for π 2 (n) to verify the Hardy Littlewood Conjecture.

4 2. Deriving the formula and some Set Theory All the twin primes except {3, 5} are of the form {6k 1, 6k+1} Let T = {(6k 1, 6k+1) k = 1, 2, 3 } Let u k = 6k 1 and let v k = 6k + 1 And define t k = (u k,v k ) Listed below are the first few members of the set T. (the composite numbers are underlined) 5,7 11,13 17,19 23,25 29,31 35,37 41,43 47,49 53,55 59,61 65,67 71,73 77,79 83,85 89, 91 95, , , , , , , , , ,151 k ± 1 mod 5 t k contains a multiple of 5 and is therefore not a pair of twin primes. Let S p = {t k t k contains a multiple of prime p} T p = T S p = {t k t k does not contain a multiple of prime p} S5 = {t4, t6, t9, t11, t 14, t16, t19 } = {(23, 25), (35, 37), (53, 55), (65, 67) } T5 = {t1, t2, t3, t5, t7, t8, t 10, t 12 } The values of k in S5 = {4, 6, 9, 11, 14, 16, 19, 21 } = {k Z + k ± 1 mod 5} The values of k in T5 = {1,2,3,5,7,8,10,12,13,15,17,18,20,22,23,25,27,28,30 } k ± 1 mod 7 t k contains a multiple of 7 The values of k in S7 = {6, 8, 13.15, 20, 22, 27, 29, 36, 38 } = {k Z + k ± 1 mod 7} The values of k in T7 = {1, 2, 3, 4, 5, 7, 9, 10, 11, 12, 14, 16, 17, 18, 19 } Since 6(2) 1= 11, k ± 2 mod 11 t k contains a multiple of 11 The values of k in S11 = {9, 13, 20, 24, 31, 35, 42, 46 } = {k Z + k ± 2 mod 11} The values of k in T11 = {1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25 } If p is prime and p = 6a ± 1, t k contains a multiple of p k ± a mod p

5 Let P = the set of all primes Let Tw = {t k = (u k,v k ) u k P and v k P, k Z + } T w = T \ S p p 5 ; by De Morgan s Law T w = p 5 T p Lemma 1: Define k p as the value of k for primes p = 6 k + 1 or p = 6k 1. and T w as the set of all twin prime pairs. Given a large integer and 6k +1 <, t k T w k ± k p (mod p) for some prime p, 5 p <. As in the sieve of Eratosthenes, we delete {k k = np ± k p for primes p = 6kp ± 1} n = {1, 2, 3 } p 5 p < Consider the set K = {k Z + k < 6}. In every interval I K such that I = {np, (n+1) p}, n Z + and p is prime, 5 p < exactly 2 values of k,(i.e. k = np+ k p and k = (n+1)p k p ),such that t k contains a multiple of p Let π 2 () = the number of primes p less than such that p + 2 is also prime. By the Brun Sieve we have: (1) π 2 () = V ( /p) + R p where R p is the error term and V= maximum prime p < Example 1: = 529, V=19 π 2 () ( /p) = Actually, π 2 () = 25 so R p 4.3

6 Let π 2 (n) = the number of primes p less than n such that p + 2 is also prime. Where V is maximum prime p < n Table 1 (π 2 (n) compared to the formula) n π 2 ( n ) n V (1 2/p) 6 5 Approx This estimate exceeds the actual number of twin prime pairs for large values of n because for some primes p T w and elements in {k : 1 k < n }, the number of elements in each of the sets {k : p 6k 6 1 } and {k: p 6k +1 } = n + 1 where is the greatest integer function, therefore some 6p composites will not be sifted out by the product formula given above. The formula can be refined by rewriting it as a two-part sieve formula that represents the application of Eratosthenes Sieve first to 6k-1 type numbers then to the 6k +1 type. See equation (2).

7 Except for 3, all lesser twin primes are of the form 6k 1. Consider the set {u k u k = 6k 1, k Z + }. Lemma 2: Given u k <, u k T w k ± k p (mod p) for some prime p, 5 p <. i.e. (u k,v k ) is not a pair of twin primes k ± k p (mod p) for some prime p, 5 p <. Out of every p elements in the set {u k }, (p prime and p 5), exactly one is a multiple of p and one precedes a 6k+1 multiple of p. If we list the elements of {u k u k = 6k 1, k Z + } and delete every u k in which k ±1(mod 5) or ±1(mod 7) or ±2(mod 11) or ±2(mod 13) or k ±3(mod 17) or ±3(mod 19) ± k p (mod p)up to p < The remaining terms are all twin primes. We use this method to find twin primes in the table below by deleting all k ±1 mod 5 or ±1 mod 7, ±2 mod 11 or ± 2 mod 13, since 13 = max p < 179 Table 2 (Values of k (not deleted) such that 6k-1 is a lesser twin prime) k u k k u k k u k The u k s that correspond to the undeleted values of k are the lesser of twin primes i.e.: 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179

8 We can demonstrate this sieve method by the following procedure: first we cross out all values of k such that k k p mod p if p 1 mod 6 ( i.e. p = 6k p 1 )and all values of k such that k k p mod p if p 1 mod 6 ( i.e. p = 6k p + 1) up to p < n so that we are left with the set {k Z + k < n and (6k-1) is prime}. 6 Table 3 Values of k such that 6k-1 is a prime (not deleted) k u k k u k k u k We then cross out the elements of the set {(k Z + k < 6 and (6k +1) is composite} i.e. {k k k p mod p if p 1 mod 6} {k k k p mod p if p 1 mod 6}. This leaves us with the set of all twin primes less than. (Table 2) This can be expressed as approximation formula that follows : (2) π 2 () π() 2 V p 2 5 π() p 1 2 V p(p 2) p 1 5 (p 1) 2 p, V= max p < By the Prime umber Theorem π () ~ ln and by Mertens Theorem: (3) V p 1 2 ~ 2e γ p ln = ln which overestimates the true ratio π() and γ = is the Euler-Mascheroni constant. [4] (Polya) By using 1 ln, which is a lower bound for π() [5] (Rosser and Schoenfeld) and a little bit of algebra, we obtain: (4) π 2 () ~ 4 C 2 ln ln, where C 2 = p(p 2) p 3 (p 1) 2 = is the twin prime constant and (5) π 2 () ~ 2C 2 (ln ) 2 which is the Hardy - Littlewood Conjecture.

9 Hardy and Littlewood [2] also conjectured a better approximation : (6) π 2 () ~ 2C (ln t) 2 dt, also based on PT Formula (4) is naturally equivalent to (3) but (7) n 1 2 (ln t) 2 dt = n (ln n) 2 (1 + 2! + 3! + ln n (ln n) 2 ), and the second factor on the right hand side is (for the values of n that we have to consider) far from negligible. [2] (Hardy and Wright ) This suggests 2C 2 (ln ) 2 < π 2 (), for large integers From equation (2) and the fact that V p is an over approximation of (because for some primes p T w, 5 p <, {k : p (6k 1) k p } p π () 2, = 6 ( p 1 p 6 ) and likewise for {k : p (6k+1) k p} ). After multiplying the right side of equation (2) by 6 and 6 we obtain: (8) π 2 () π() C 6 π() 2 where C = p(p 2) 2<p< (p 1) 2 which includes lim ( (p 2) 6(p 1) )( (p 1) p (p 1) 6p ),for some primes p, 5 p <. (9) π 2 () 2C [π()]2. lim C = C 2 π () ~ ln (10) π 2 () ~ 2C 2 (ln ) 2 As shown by Rosser and Schoenfeld [5], ln < π ( ) 17 which gives us : (11) 2C 2 (ln ) 2 < 2C [π()]2 2 for large enough values of.

10 Table 4 (Values of π 2 (n) compared to logarithmic integral and ratio formulas) [1] (Caldwell) n π 2 (n) 2C 2 li 2 (n) 2C 2 (ln n) n Let W (n) = 2C 2 [π(n)]2 n Table 5 (limit W (n) π 2 (n) approaching 1 as n increases) n π (n) W(n) W(n) π 2 (n)

11 Let t k = (6k 1, 6k + 1) and T w = the set of all twin prime pairs. The occurrence of twin primes may be summarized as follows: k > 3 and primes p, t k T w k 0, 2 or 3 (mod 5) k ± k p (mod p) p > 5 Where k p is the value of k for the primes p = 6k +1 or p = 6k 1

12 References: 1.Caldwell, C.: An Amazing Prime Heuristic utm.edu/staff/caldwell/preprints/heuristics.pdf 2. Halberstam, H., Richert, H. E.: Sieve Methods. Academic Press (1974) 3. Hardy, G. H., Littlewood, J. E.: Some Problems of Partitio umerorum III: On the Expression of a umber as a Sum of Primes. Acta Mathematica 44, (1923) pp Lagarias, J.: Euler s Constant: Euler s Work and Modern Developments. Bulletin of the American Mathematical Society 50 (4), (2013) pp Polya, G.: Heuristic Reasoning in the Theory of umbers. The American Mathematical Monthly 66 (5), (1959) p Rosser, J. Barkley, Schoenfeld, Lowell: Approximate formulas for some functions of prime numbers. Illinois J. Math. 6, (1962) p Wolf, M.: Some Conjectures on the Gaps between Consecutive Primes (1998) p.5